of 30
Chapter 9
RADICAL EXPRESSIONS,
FUNCTIONS, AND EQUATIONS
and Functions
variable appears in a radicand.
24. 55
26.
(
)
781 73 73 21===
3
−=− =−=−
⎟⎟
⎜⎜
⎟⎟
⎜⎜
()
()
(
)
()
4
54.
(
)
()
727 7 3
58.
(
)
ab ab ab==
158
60.
(
)
(
)
2225
45
f=+
=+
(
)
64.
(
)
(
)
f=+
83
=−⋅
3
(
)
3 3 4 1.73 4 2.27g=−≈ −=−
()
x
()
2gx x=−
()
,xy
(
)
72. 55
(
)
82.
()
9gx x=+
(
)
3
(
)
339123.46g=+= ≈
(
)
159
The length of each side of the box is 12 in.
()
The length of each side changes by a
will be plotted at the origin or in the first
quadrant of the coordinate plane. So only
3. a. Answers may vary.
positive numbers. However, only one
counterexample is needed to prove an
(
)
8. 12
81 81
9
−=−
=−
14.
(
)
()
2
25 25
5
nn
n
n
=
=
(
)
2
y
=−
18.
(
)
()
5
44
2
=
=
=
160
()
22.
(
)
(
)
()
()
32 32
xx
−=−
⎜⎟
⎝⎠
()
2
1
4
1
16
=
=
9
3
8
y
=
28.
()
4
888
2
⋅=
=
30.
13
3
=
1
4
⎛⎞
=
34.
(
)
()()
16
81 81
=
36. 56 12 56 12
66
aa a
⋅=
38.
(
)
3
43 433
4
1
xx
x
−−⋅
−
=
=
()
()
3
2
3
3
y
=
7
14
33
44
uu
v
⎛⎞
161
()
()
()
26
216
nn
⎜⎟
=−
()
3
12
x
()
=
52.
8
16
a
⎜⎟
⎜
⎟⎟
⎜⎜
⎟
⎟
⎜⎝⎠
⎝⎠
54.
33
32
yy
x
⎜⎟
⎜⎟
⎟
⎜
⎝⎠
()
(
)
(
)
(
)
(
)
35.74 0.62 30 35.75 5 0.43 30 5
24
+− +
≈
It takes Mercury about 86 Earth days
2. Answers may vary.
3. a.
()
b.
(
)
()
(
)
22
912 4 32
aabb ab
++ =+
(
)
(
)
22
3
3
xy xy
=+ = +
()
162
Expressions
4. The distance between two points
(
)
11
16
6
n
18. 63
20. 84
nnnn n n
⋅=⋅= =
a
a
57 2 35 2
=⋅ =
46. 32 2
52. 79 68
54. 33 33
58. 28 6 22
250 125 2
pq q pq
=⋅
Section 9.3 Simplifying Radical Expressions
163
13
13
6
6
q
q===
32
5
5
n
n
==
3
3
pq
pq
64 8
64
==
100 10
100
==
8
xx
==
144 12
144
==
6
36 6
36
qq
q
==
4
==
==
49 7
49
rs rs
rs
()
()
()
22
27 58
81 9
910
3 10 units
=−−+−
=+
=⋅
=
()( )
()()
()()
21 21
22
22
19 82
10 10
100 100
100 2
=−− +−−
=− +−
=+
=⋅
()( )
()
()()
21 21
22
43
25
5 units
dxx yy=−+−
=− +−
=
=
7
7
x
x==
36
63
n
n
== ==
2224
=⋅=
93
81 81
bb
bb
===
Chapter 9 Radical Expressions, Functions, and Equations
164
()
()
22
25 2
=⋅
16
110. 0.75 8
6.3639
px
=−
=
≈
362,500
2500 145
=
=⋅
(
)
4, 6 .−−
()
()
22
46
16 36
=+
=+
1. nais in its simplest form if a contains
2. The expressions are equivalent.
3. a.
()
()()()
(
)
1
111
xxx
=++−
=
()
()
2
x
+
()
3
x
+
165
of Radical Expressions
Exercises
4. The Pythagorean theorem can be expressed
(
)
12.
(
)
16.
(
)
813 3 8133
aaa a
− + =−+
18.
(
)
11 15 11 1 15
xx x x
−−+ =−−+
35 45
=+
()
25 3
33
=−
=−
32.
2
16 2 49 2
yy yy
=⋅− ⋅
36.
2
38.
42
69
a b ab b a ab
=⋅−⋅
166
40.
()
27 4 64 3 4 4
rrr rrr
−+ =−+
42.
55
522 22
55
10 2 4 64
10 2 4 32 2
10 2 4 2 2
xy xy
xy xy
yxy xy
+
=+⋅
=⋅+⋅
44.
128 98 225
−+
()
33
2
63 123 4
=− + ⋅ −
=− + −
48.
() ()
()
42
42
2
108 147
fx gx x x x
+=− +
()
2
13 3
x
=−
()
()
3
22
12 6
xx
=−
()
52.
ab ab b ab ab
−=⋅−⋅
56.
()
()
24
2
() 2 63 9 7
29 79 7
15 7
fx gx x x x
xx x
x
+=− −
=− ⋅ − ⋅
=−
()
2
23 79 7
xx x
=− ⋅ +
Section 9.4 Addition and Subtraction of Radical Expressions
167
58.
(
)
(
)
1.5 32, 000 1.5 18,000
10 30 54.8
−
=≈
60. side = 52, 900 230=
Perimeter = 4 side
×
62. a. 22
22
200 160
40,000 25,600
14,400
120
dlh=−
=−
=−
=
=
120 96.8 23.2 m.−=
Mindstretchers
1.
(
)
()
48
gx x
=+
(
)
(
)
2522545210
h
=+==⋅=
b.
(
)
(
)
(
)
(
)
(
)
226410
222
fg
fgh
+=+=
+=
c. Answers may vary. For 3 :x=
e.
(
)
() 9 18 4 8
fx gx x x
+=+++
(
)
(
)
(
)
(
)
0.bb+= If 0a≥ and 0,b= then
00aaa+= += and 0.aa+=
168
9.5 Multiplication and Division
of Radical Expressions
Exercises
6. To rationalize a denominator with two
12.
(
)
(
)
33 3
3
24 414 24414
82 7
=⋅ ⋅
=⋅
14.
(
)
(
)
412 36 4312 6
12 36 2
yy yy
y
=⋅ ⋅
=⋅
16.
(
)
(
)
()
23
6565
30
30
pq pq pq pq
pq
pq
−−=−⋅−⋅
=
=
48
4
24
16 25
225
xy x
xy x
=− ⋅
=−
(
)
()
627 122
=− ⋅ + ⋅
26.
(
)
8875 88785
−−=−⋅+
28.
(
)
33
yyy yy yy
−=⋅−⋅
30.
(
)
33
22
274 572 4 2 5
72 4 2 5
xx xxx
xx x
−= ⋅ − ⋅
=⋅−⋅
169
()
34.
(
)
(
)
()
()
45 4 5 1
21 4 5
=− ⋅ + −
=− +
(
)
(
)
()
38.
(
)
(
)
()
40.
(
)
(
)
(
)
(
)
8383 8 3
+−=−
42.
44.
(
)
(
)
(
)
(
)
77 7
7
qp qp q p
qp
−+=−
=−
46.
(
)
(
)
(
)
yy y
+− ++ = + −
22
2
(
)
(
)
(
)
96 6 6
663
xx
xx
=− −+−
=− −+
2
16
16
a
=
55
108 108
3
yy yy
170
33
164 2
x
=
62. 222
49 7
49
68.
== =
70. 2
62
ppqpq
82.
33 3
222
3
333
23
29 18 18
3
3927
ab ab ab
b
bbb
⋅= =
86. 52
222
qqq
=⋅
2
72 2
vv v
90.
81 81
rr
=
327
r
()
()
xxy
x
+
=
171
()
()
2
418 12
6
43 2 2 3
6
26 2 3
6
62 3
nn
n
nn
n
nn
n
nn
−
=
⋅−
=
−
=
−
33
323
a
=
()
35 3
4
−
=
(
)
()()
()
()
()
12 3 6
12 3 6
36
12 3 6
3
43 6
43 46
+
+
=−
+
=−
=− +
=− −
()
()
()
()
()
96 3
36 3
96 3
312
36 3
12
y
y
y
y
y
y
+
=−
+
=−
+
=−
()
()
2
ab b
+
()
()()
22
15
5
55
5
nn
n
nnn
n
−+
=
−
+−−
=−
() ()
()()
22
22
2
2
pqp q
ppqq
pq
ppqq
pq
−+
++
=
−
++
=−
172
() ()
()()
22
22
416 4
4
416 4
xxyxyy
xy
xxyxyy
+++
=
−
+++
=−
114.
()
33
3
427 4
43 4
xxx
xx x
=⋅
=⋅
2
33
49
43 2
42
x
x
=⋅
116.
(
)
(
)
(
)
(
)
()
24
68
fxgx x x
xx
⋅=+ +
=+ +
()
()
2
16
x
=
−
()
2
2
320 50
10
34 5 252
10
rr
r
rr
r
−
=
⋅− ⋅
=
120.
(
)
() ()
33
33
9418 715
4162 7135
4276 7275
=− +
=− ⋅ + ⋅
122.
(
)
(
)
(
)
63 63 6 3
xx x
++ +− = + −
124.
()
()
3
81
55
95
m
nn
mn
=
126.
() ()
()
2
1
27 14
2
72
fxgx x x x
xx
⋅= ⋅
=
173
126. (continued)
2
414
12
422
42
2
22
xx
x
x
x
=⋅
=⋅ ⋅
=
=
Radius is half the diagonal. 2
s
r=
()
2
2
22
2
4
2
42
s
ss
π
π
π
⎛⎞
⎟
⎜⎟
⎜⎟
⎜⎟
=⎜⎟
⎜⎟
⎟
⎜⎟
⎜
⎝⎠
⎛⎞
⎟
⎜⎟
==
⎜⎟
⎜⎟
⎟
⎜
⎝⎠
()
165
rh
=
132.
333
23
PPP
1.
(
)
(
)
()()
()
()
?
?
322 6322 10
9122 818122 10
9 8 18 1 12 2 12 2 0
00
−−−+=
−+−++=
+− + +− + =
=
()
x
x
()
xx
c.
()
()
()
()
11
11
11
11
1
xx
xx
x
+
−+
−
+−
⎛⎞ ⎛⎞
=−+
++
111
2
21
xxx
x
−++−
−
174
24yx=−
b. The domain of the expression 2x+ is
2,x≥− and the domain of the expression
2x− is 2.x≥ Since the expression 2x−
the product 22xx+⋅ − is restricted to the
domain of the expression 2.x− The domain of
the expression 24x− is the set of all real
numbers x such that 240,x−≥ 2,x≤−
not the same, the graphs of their equations are
different.
Equations
Exercises
()
2100
50
a
a
a
=
=
=
()
716
23
x
x
−=
=
6.
()
415
4125
424
6
x
x
x
x
+=
+=
=
=
8.
()
12 5 3
12 5 27
515
3
x
x
x
x
−=
−=
−=
=−
12.
()
87
225
x
x
−=
=
=
14.
()
3
15 13
8
n
n
+=
=−
Section 9.6 Solving Radical Equations
175
()
=
()
549
5481
585
n
n
n
−=
−=
=
20. 21 3 15 24
x
−−=
()()
45 27
4527
nn
nn
+= +
+= +
24.
()()
6113 7
352
xx
x
−= −
−=−
()()
29314
xx
+= +
28.
(
)
(
)
2
3100
nn
+−=
Check 5.n=−
?
25 8 2 15
17 17
−= +
=
() ()
28232
−= −
30.
()
()
33
2
33
10 9 8 0
10 9 8
pp
pp
++ +=
+=−+
Check 6.p=−
00
=
176
Check 3.p=−
() ()
?
2
33
310 9380
−++ −+=
32.
()
2
32
320
yy
yy
−=
−+=
?
?
321
−=
?
?
622
−=
()
2
2
619
9610
aa
aa
−=
−+=
1
Check .
3
a=
00
=
36.
()
2
25 1
25 21
xx
xxx
+=+
+= + +
Check 2.x=−
()
?
22 5 21
−+=−+
()
?
22 5 2 1
4521
+=+
+=+
177
38.
()
2
5932
594 129
xx
xxx
++=
+= − +
Check 0.x=
?
930
+=
4
?
17 17
5932
44
⎛⎞ ⎛⎞
⎟⎟
⎜⎜
++=
⎟⎟
⎜⎜
⎟⎟
⎜⎜
⎝⎠ ⎝⎠
()()
()
(
)
(
)
123
1962 2
xx
xxx
−+ =
−= − +
Check 2.x=
()
?
21 22 3
−+ =
Check 50.x=
()
?
?
50 1 2 50 3
49 100 3
−+ =
+=
178
(
)
(
)
22
46 2
rr
−+ =−−
()
96
r
=+
?
436 32
43 1
11
−+=−−
−=−
≠−
()()
()
14 3 6 3 9
22 6 3
aa a
aa
−=++ ++
−= +
Check 13.a=
?
14 13 13 3 3
−= ++
Check 2.a=−
()
?
14 2 2 3 3
−− = − + +
The solution is −2.
()()
()
2
21221131
xx x
−+ −+ = +
15
xx
==
Check 1.x=
11 4
+=
?
?
Check 5.
91 16
314
x=
+=
+=
179
()
323
60 20
xx
=−=
?
Check 0.
x=
?
50.
()
2
2
7423
749 124
09 5
xxx
xx
+= + +
=+
52.
()
3
13 9
913
64
p
p
p
+=
=−
=−
()
()
2
2
11 7 3
3180
aa
aa
−=−
+−=
() ()
Check 6.
611736
a=−
−−=−−
Check 3.
911 79
−= −
()
2
22
2
18 128
324 128
h
h
=+
=+
2
64
vd
v
=
b.
64 64
d== =
100 ft
60.
()
2
2
10 6
10 36
100 36
64
A
A
A
A
=+
=+
=+
=
180
1. Equivalent equations have the same
solutions. Raising each side of the radical
solutions. Since the equations do not
()
2
196
xxx
−= − +
?
?
51 35
42
22
−=−
=−
≠−
?
?
2132
11
11
−=−
=
=
3. a. 292x−=
()
92
x
⎟
⎜−=
⎟
⎜⎟
⎜
⎝⎠
()
?
2
592
−−=
Check 5.x=
()
?
2
?
?
592
25 9 2
−=
−=
181
()
2
3
21 4
x
+=
()
7
Check .
2
x=
22
=
9.7 Complex Numbers
2. A complex number is any number that
can be written in the form ,abi+ where
12. 10 10i−=
18. 21 2i−−=−
22. 11
22
116 2
i
=− ⋅
24.
(
)
26.
(
)
(
)
182
30.
(
)
(
)
34.
(
)
(
)
()
36.
(
)
(
)
()
()
55 1
=−
42.
()
44.
()
25 2 10 4
iiii
−=−
46.
(
)
44 8 16 32
ii ii
−+=−−
48.
(
)
(
)
ii
−+−=+
() ()
317 20
i
=+ −
2
58.
(
)
(
)
36 63
ii
−− −
(
)
(
)
(
)
(
)
(
)
183
(
)
(
)
(
)
(
)
(
)
2
924 16
ii
=− +
()()
()
22
2 3 43 438
68
iii
=+−−
=+
68.
Complex Conjugate Product
Number
70.
Number
72.
Complex Conjugate Product
Number
74.
Number
(
)
(
)
(
)
22
09 09 9 81 81
iiii−+−=−=
78.
23 2323
13
23
13 13
iii
i
=⋅
++−
=− +
()
()()
33
10 10
ii
i
=−+
=+
4
i
−
=
15
i
=
86.
2
222
12 4
iii
ii
=⋅
−
−−
184
88.
333
9
iii
i
=⋅
−−+
537
34
i
i
−
=
92.
4242
44
16 4
ii
=⋅
+−
+−
(
)
()
33
31 3
10
i
i
ii
+
=−
==
(
)
98.
(
)
(
)
1iiiiiii== ==
()
71
i= = ⋅= ⋅= ⋅=
104.
(
)
()
108.
222
42
5
iii
i
=⋅
−−+
−−
=
110.
(
)
6 5 12 30 72
ii ii
−−=−+
114.
()()
8593
VIZ
ii
=
=+ +
116.
(
)
()()
252
36 2 36 52
fz z i
fi ii
=++
−+ = −+ + +
185
(
)
3.
() ()
()()
?
2
?
?
12 22 20
12 122 20
112 2 2 2 0
00 True
ii i
ii
ii
++−−+=
+−−−+=
−− + + − =
=